Transition Probabilities - Markov Process

Question

Suppose there is a box with $N$ balls. Each ball is coloured either red or blue. In each time period, one ball is chosen at random from the box and with probability $\frac{1}{2}$ replaced with the other colour; or the ball is returned to the box. Let $X_{n}$ denote the number of red balls after $n$ picks.

A) Determine the transition probabilities: $$p_{ij}, \forall{i,j}=0,1,....,N$$

Any help would be much appreciated thank you.

Answer 1

Hints:

1. If you have $x$ red balls after $n$ picks, what is the probability that your $(n+1)$th pick is red?

2. or blue?

3. What happens to the number of red balls in each of these cases?

4. If you have $x$ red balls after $n$ picks, what are the possible numbers of red balls after $n+1$ picks?

5. What are the probabilities that you transition to each of these possibilities?